MoL-2010-03:
Vlek, Charlotte
(2010)
*Definability in the Degrees of Randomness.*
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## Abstract

A set or sequence is random when the pre~x-free Kolmogorov complexity

of its initial segments is relatively high: equal to the length of the

segment (up to a constant). Using Kolmogorov complexity of initial

segments, we can not only de~ne when a set is random, but we can also

compare which of two sets is more random. We say that a set A is

K-below (or K-reduces to) a set B if $K(A \upharpoonright n ) <=^+

K(B \upharpoonright n)$ for all $n$. This reducibility gives the

structure of the K-degrees. The sets in the lowest degree are called

K-trivial sets.

This thesis studies arithmetical de~nability in the K-degrees. The

main result we present, is the construction of a non-K-trivial

\Delta^0_2 set that does not bound any non-K-trivial set in a given

Delta^0_2 family of sets. This implies that there is a non-K-trivial

Delta^0_2 set that does not bound any non-K-trivial c.e. set.

Furthermore, this result shows a structural di~erence between the

K-degrees and the LK-degrees.

Similar to the above result, we also show that for all n > 1 there is

a non-K-trivial \Sigma^0_n set that does not bound any non-K-trivial

Delta^0_n set. We present the construction for the particular case of

n = 2, and we show that this speci~c \Sigma^0_2 set forms a minimal

pair in the K-degrees with any non-K-trivial c.e. set. This improves

on the lowest complexity known so far for minimal pairs in the

K-degrees.

Finally, we investigate the possibility of constructing a minimal pair

in the K-degrees via gap functions for K-triviality. We show that no

unbounded non-decreasing Delta^0_2 gap function can exist, thus

showing that this method is not suitable for constructing a Delta^0_2

minimal pair in the K-degrees.

Item Type: | Report |
---|---|

Report Nr: | MoL-2010-03 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2010 |

Date Deposited: | 12 Oct 2016 14:38 |

Last Modified: | 12 Oct 2016 14:38 |

URI: | https://eprints.illc.uva.nl/id/eprint/828 |

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